Probabilistic analayses were performed to assess the effects of the time variant exposure to carcinogen on cancer risk estimates. Equations were formulated from several statistical models of cancer that allow one to determine the effects on risk estimates of (1) a time-varying toxic agent concentration and of (2) a migrating exposed population. The incidence models examined included the one-hit, multihit, and Weibull models. Present results show that the same risk estimates are obtained from the one-hit and multihit models when either the time-averaged dose or the fully time-dependent dose is used. The Weibull model requires that time-dependence be carried through the risk estimation procedure or else order of magnitude errors may occur (such as for the 30 year arsenic risk estimate when exposure actually occurs only over a third of this time). Migration was described by a first order partial differential flow model and was coupled to the incidence models via a residence time distribution formalism. Failure to include such a description of population migration results in both multihit and Weibull models overpredicting estimated risk, e.g. a Weibull model applied to arsenite toxicity will overpredict the incidence of skin cancer by ten-fold if an out-migration rate of 12% per year is ignored. In addition, the multistage model for an arbitrary number of stages has been solved for constant concentration exposure. In turn, this multistage model along with multihit and Weibull models, was used to recast the form of cancer incidence expected at the cellular level to that expected at the organ level, the correct form to use when fitting to animal dose-response data.